What is the median value of the set R, if for every term in

Question

What is the median value of the set R, if for every term in the set,  R_{n} = R_{n–1} + 3 ?

(1) The first term of set R is 15.
(2) The mean of set R is 36.

KarishmaB · Accepted Answer

There are certain issues with the question. A set does not have elements in a sequence so there is no question of having R n  and R (n-1) 
It needs to be something like this: Elements of a set are arranged in increasing order and it is observed that except for the first element, every element is 3 more than the previous element.

Anyway, I assume that the intent of the question is this.

In that case, notice that this is an arithmetic progression (numbers are evenly spaced).

In an AP, mean = median (since both are the middle term). Hence statement 2 alone is sufficient. 
In statement 1, you need to know the total number of elements too to find the median.

penguinman · Answer

1. Doesn't say anything about the how many elements are in the set. Insuff
2. This is an evenly spaced set by multiples of 3, the median and mean for an evenly spaced set are equal. Suff
Answer: B

As a side note, can we also assume that there are an odd number of elements because the mean = median is an integer?

penguinman · Answer

VeritasPerepKarishma, is this just the definition of a set that you are referring to, can you provide a resource to look this up? I was under the impression that a set can contain a sequence.

KarishmaB · Answer

A set is a collection of objects, not a sequence. Check here: https://en.wikipedia.org/wiki/Set_(mathematics)

penguinman · Answer

Ok, but from https://en.wikipedia.org/wiki/Sequence, a sequence is an ordered list of objects. So why couldn't a set contain a sequence? Isn't the OQ just saying that a set contains a sequence abiding by the equation provided above? I guess I am arguing technicalities but I just want to be clear.

Thanks!

KarishmaB · Answer

The question does not talk about sequences. It talks about sets. When one says 'set', you think of a group of numbers, not numbers in a particular sequence. It is confusing to someone who is reading it for the first time. You wonder about Rn and R(n-1) and what they mean. You have to guess the intent of the question. GMAT questions do not do that.

Bluelagoon · Answer

8. What is the median value of the set R, if for every term in the set, Rn = Rn–1 + 3?
(1) The first term of set R is 15. (2) The mean of set R is 36.

For an AP, mean= median. Hence B is sufficient.
A doesn't give any idea about the number of terms and hence we cant find median.

mejia401 · Answer

Great insight, VeritasPrepKarishma

It´s similar logic on how:  all squares are parallelograms, but not all parallelograms are squares . Accordingly, all sequences are sets, but not all sets are sequences (excluding sequences with no end, infinity, of course)

KarishmaB · Answer

Actually a 'set' is a collection of objects with no order. A 'sequence' is an ordered list of objects. They are two different things. The only reason I framed the question that way was to keep the original framing involving sets.

hsbinfy · Answer

B it is .

i would like to add some more to it.
if the AP is consecutive odd even odd evn,which is the case here....then the median=mean in the above case.

Suppose the AP above was r(n)=r(n-1) +4 then ans would not be B. We would require some more information

I hope you get what i am saying.

lastshot · Answer

Hi Karishma !!!

Will you please elaborate with an example how in AP or evenly spaced sequence Mean=Median --> just want to know reason or theory behind it

Thanks in advance

Regards
Last Shot :2gunfire: :2gunfire:

KarishmaB · Answer

What is the mean of 43, 44, 45, 46, 47?

Arithmetic mean is the number that can represent/replace all the numbers of the sequence. Notice in this sequence, 44 is one less than 45 and 46 is one more than 45. So essentially, two 45s can replace both 44 and 46. Similarly, 43 is 2 less than 45 and 47 is 2 more than 45 so two 45s can replace both these numbers too.

The sequence is essentially 45, 45, 45, 45, 45.

Hence, the arithmetic mean of this sequence must be 45! (If you have doubts, you can calculate and find out.)

It makes sense, doesn’t it? The middle number in the sequence of consecutive positive integers will be the mean. The deviations of all numbers to the left of the middle number will balance out the deviations of all the numbers to the right of the middle number.

Once again, what is the mean of 192, 193, 194, 195, 196, 197, 198?

It is 195 since it is the middle number!

Ok, what about 192, 193, 194, 195, 196, 197? What is the mean in this case? There is no middle number here since there are 6 numbers. The mean here will be the middle of the two middle numbers which is 194.5 (the middle of the third and the fourth number). It doesn’t matter that 194.5 is not a part of this list. If you think about it, arithmetic mean of some numbers needn’t be one of the numbers.

What about 71, 73, 75, 77, 79? What will be the mean in this case? Even though these numbers are not consecutive integers, the difference between two adjacent numbers in the list is the same (it is an arithmetic progression). So the deviations of the numbers on the left of the middle number will cancel out the deviations of the numbers on the right of the middle number (71 is 4 less than 75 and 79 is 4 more than 75. 73 is 2 less than 75 and 77 is 2 more than 75). Hence, the mean here will be 75 (just like our first example).

Just to re-inforce:

102, 106, 110 –> Mean = 106

102, 106, 110, 114 -> Mean = 108 (Middle of the second and third numbers)

Now think, what is the median? It is the middle number! Hence, in an AP, mean = median.

eddyki · Answer

Hello guys!
One question.Can a set start with a negative number? Because:
If the set starts with -3
we get for the mean 36 and the median is 34,5

1 -3
2 0
3 3
4 6
5 9
6 12
7 15
8 18
9 21
10 24
11 27
12 30
13 33
14 36
15 39
16 42
17 45
18 48
19 51
20 54
21 57
22 60
23 63
24 66
25 69
26 72
27 75

thanks in advance!

EMPOWERgmatRichC · Answer

Hi eddyki,

The answer to your immediate question is YES - a sequence CAN start with a negative term.

HOWEVER, you have made a mistake with your example:

Your group of numbers includes 27 terms, so the MEDIAN is the 14th term (NOT the average of the 13th and 14th terms).

Here, the mean AND the median are both 36.

GMAT assassins aren't born, they're made,
Rich

gmatlbs · Answer

In the original question , in place of "median" if the question is about "SD" , will the answer be "C"?
Please suggest.

"What is the SD of the set R, if for every term in the set, Rn = Rn–1 + 3?"
(1) The first term of set R is 15.
(2) The mean of set R is 36

Approach - 
(1) The first term of set R is 15. Insuff - Set is evenly spaced but #of terms is not know.
(2) The mean of set R is 36. Insuff - Same issue as above.

Combining the two , set becomes :
15 , ...( 18 to 33) ...36...(39 to 54)...57.
# of terms = 15 
Difference between the terms and mean is know. Hence , Sufficient C.

stonecold · Answer

Excellent Question.
Testing all our concepts of Evenly spaced sets.

In any evenly spaced set => Mean = Median = Average of the first and the last term.

So we basically need the mean to get the median.
Statement 1=>
No clue of the number of terms => Not sufficient

Statement 2=>
Sufficient as mean = median 
Hence B

gagan0303 · Answer

Bunuel  Can u please help me with this one. How is only B sufficient?

Gagan

achloes · Answer

The question stem tells us that the elements in the set are evenly spaced by a common difference of 3. This also means that the median will equal the mean.

S1. We are given the first term but no further info. The median or mean also requires the last term of the set: (first term + last term)/2

INSUFFICIENT

S2. We are given the mean, which means we know the median value

SUFFICIENT

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